5. Three-Dimensional Geometry

Cylinder

A cylinder in three-dimensional coordinate geometry is generated by a moving straight line (generatrix) that remains parallel to a fixed direction and intersects a given fixed curve (guiding curve).

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Equation of a Right Circular Cylinder

Let the axis of a right circular cylinder be given by the symmetric equations \frac{x-x_1}{l} = \frac{y-y_1}{m} = \frac{z-z_1}{n} where l^2+m^2+n^2=1$, and let $r$ be its constant radius. The perpendicular distance from any point $P(x,y,z)$ on the surface of the cylinder to the axis is identically equal to $r$.

📐 Geometric Interpretation

The surface is traced in 3D space by revolving a line parallel to the axis at a fixed constant perpendicular distance $r$. Cross-sections perpendicular to the axis are congruent circles of radius $r$.

⚙️ Algebraic & Structural Meaning

Translates spatial geometric locus conditions into a second-degree algebraic equation in three variables $(x, y, z)$ lacking constant term and mixed linear-quadratic anomalies when properly centered.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Generatrix Direction Cosines	`l^2 + m^2 + n^2 = 1 (Normalisation)`	Strictly Mandatory / Necessary	If direction ratios are not normalized when computing perpendicular distance via cross product, distance formula yields scaled incorrect radii.
Guiding Curve Specification	`Intersection of plane and surface`	Strictly Mandatory	Failure to define a unique base guiding curve leaves the axial extent or cross-sectional shape undetermined.
Radius Reality Condition	`r > 0`	Essential for real surface	If $r^2 < 0$, the cylinder degenerates into an imaginary cylinder with real axis; if $r = 0$, it collapses into a degenerate line cylinder (axis itself).

Useful results and shortcuts

Use these identities and results when their conditions apply:

Right Circular Cylinder with Axis along Coordinate Axes

$$x^2 + y^2 = r^2 \quad (\text{Axis } z\text{-axis})$$

Rule: If the axis coincides with a coordinate axis, the equation simply drops the coordinate corresponding to the axis and equates the sum of squares of the other two coordinates to $r^2$.

Application: undefined

Radius from General Equation of Right Circular Cylinder

$$a(x^2+y^2+z^2) + 2ux + 2vy + 2wz + d = 0 \implies \text{Conditions: } u^2+v^2+w^2 > ad, \text{ and coefficients of } x^2, y^2, z^2 \text{ must be equal}.$$

Rule: Equate coefficients of $x^2, y^2, z^2$ first. If unequal, it cannot represent a right circular cylinder. Use symmetry to extract radius rapidly.

Cylinder with Given Generatrix and Guiding Curve

$$\text{Eliminate } (a, b) \text{ from } \frac{x-x_1}{l} = \frac{y-y_1}{m} = \frac{z-z_1}{n} \text{ and guiding curve } f(a,b)=0, g(a,b)=0$$

Rule: Use point-direction form of the moving generator line passing through $(a,b,0)$ or guiding curve points, then eliminate parameters.

Application: Direct locus problems in UP PGT Mathematics.

Enveloping Cylinder Equation

$$S S_1 = T^2$$

Rule: For a sphere $S = 0$ and parallel rays with direction ratios $(l, m, n)$, the enveloping cylinder is found using standard tangent conditions or polar plane relations.

Axis Length and Foot of Perpendicular

$$d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}}$$

Rule: Distance from external point to axis gives radius $r$ instantly when constructing cylinder through a given point parallel to a line.

Rapid Option Elimination via Homogeneity

$$Degree of second degree terms = 2$$

Rule: A cylinder equation representing lines parallel to a fixed line has its highest degree homogeneous part factorizable into linear factors corresponding to parallel planes or degenerate forms.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Unequal Coefficients of Quadratic Terms

Common Pitfall: Assuming any second-degree equation in $x, y, z$ with $x^2, y^2, z^2$ terms represents a right circular cylinder.

Counterexample: $2x^2 + 3y^2 + 2z^2 = 5$

Why it fails: The coefficients of $x^2$ and $y^2$ or $z^2$ must be strictly equal for a right circular cylinder. Here it represents an ellipsoid.

🛡️ Exam Defense: Check coefficient equality of $x^2, y^2, z^2$ before testing radius conditions.

⚠️ COMMON ERROR #02

Ignoring Axis Direction Cosine Normalization

Common Pitfall: Using raw direction ratios $(l, m, n)$ instead of actual direction cosines when calculating perpendicular distance $r$.

Counterexample: \text{Using } l=2, m=3, n=6 \text{ without dividing by } \sqrt{2^2+3^2+6^2}=7

Why it fails: Results in dimensional scaling error, yielding incorrect radius calculations.

🛡️ Exam Defense: Always ensure $l^2+m^2+n^2 = 1$ when applying perpendicular distance formulas involving line directions.

Comparison

Concept / Surface	General Equation Form	Key Parameter / Axis Condition	Cross-Sectional Shape	Exam Identification Clue
Right Circular Cylinder	x^2+y^2+z^2 + 2ux + ... = 0 (with equal quadratic coefficients)	Axis parallel to fixed line, radius $r$	Circle	Equal coefficients of squared terms and vanishing Jacobian conditions
General Cylinder	F(x, y, z, l, m, n) = 0	Generatrix parallel to fixed vector $(l,m,n)$	Shape of guiding curve	Absence of one variable or explicit elimination form
Enveloping Cylinder	S \cdot S_1 = T^2	Envelope of lines parallel to a given direction touching a sphere/conicoid	Conic / Similar to base surface	Tangency condition to a given quadric surface
Right Circular Cone	ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0	Vertex at origin, semi-vertical angle $\theta$	Circle (perpendicular to axis)	Homogeneous second-degree equation

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Find the equation of the right circular cylinder whose axis is the line $\frac{x}{1} = \frac{y}{2} = \frac{z}{2}$ and whose radius is $3$.

(A) $4x^2 + y^2 + z^2 - 4xy - 4yz - 2dx + \dots = 45$(B) $5x^2 + 5y^2 + 5z^2 - 4xy - 4yz - 4zx = 27$(C) $x^2 + y^2 + z^2 = 9$(D) $3x^2 + 4y^2 = 25$

📚 Method A: Formal Textbook Derivation

Step 1: Let $P(x,y,z)$ be any point on the cylinder. Step 2: Compute the length of the projection of vector joining origin to $P$ on the axis line. Step 3: Use Pythagoras theorem: $r^2 = OP^2 - \text{(projection)}^2$. Step 4: Simplify to get $5(x^2+y^2+z^2) - 4(xy+yz+zx) = 27$.

⚡ Method B: 25-Second Exam Shortcut

Apply distance from point to line formula: $d^2 = \frac{(y_2-y_1)^2 + \dots}{\sum l^2} = 3^2 = 9$, expand and match coefficients.

💡 Tip: Be extremely careful with cross-product expansions in distance formulas to avoid sign slips.

CANONICAL PROBLEM 02 Dual-Method Breakdown

The radius of the circular cylinder $x^2 + y^2 + z^2 - yz - zx - xy = 1$ whose axis is given by $x = y = z$ is:

(A) $1$(B) $\sqrt{\frac{2}{3}}$(C) $\sqrt{2}$(D) $\frac{1}{\sqrt{2}}$

📚 Method A: Formal Textbook Derivation

Step 1: Identify the axis line $x=y=z$ with direction ratios $(1,1,1)$. Step 2: Normalize direction cosines to $\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$. Step 3: Compute distance from any point on surface to axis. Step 4: Alternatively, compare with standard right circular cylinder forms to find $r=\sqrt{2}$.

⚡ Method B: 25-Second Exam Shortcut

Recognize that for symmetric axis $x=y=z$, the quadratic form reduces cleanly yielding $r^2 = 2$, hence $r=\sqrt{2}$.

💡 Tip: Symmetric axes simplify cross-product terms significantly in examination settings.

Practice questions

Choose an answer to check it and see the solution:

The equation $x^2 + 2y^2 + z^2 - 2xy - 2yz = 7$ represents:

A A sphere

B A right circular cylinder

C A cone

D An ellipsoid

Derivation & Analysis:

Checking the determinant of the second-degree homogeneous part and evaluating rank and cofactor conditions confirms it represents a cylinder.

What is the radius of the right circular cylinder $x^2 + y^2 = 9$?

A $3$

B $9$

C $6$

D $3\sqrt{2}$

Derivation & Analysis:

The equation $x^2 + y^2 = r^2$ represents a right circular cylinder along the z-axis with radius $r = \sqrt{9} = 3$.

The axis of the right circular cylinder given by $x^2 + y^2 + z^2 - 2xy - 2yz - 2zx + 9 = 0$ is parallel to:

A x-axis

B y-axis

C The line $x = y = z$

D z-axis

Derivation & Analysis:

Symmetric appearance of cross terms $-2xy - 2yz - 2zx$ indicates the axis is aligned symmetrically along the line $x = y = z$.

If the guiding curve of a cylinder is $x^2 + y^2 = a^2, z = 0$ and generators are parallel to the z-axis, its equation is:

A $x^2 + y^2 = a^2$

B $x^2 + z^2 = a^2$

C $y^2 + z^2 = a^2$

D $x^2 + y^2 + z^2 = a^2$

Derivation & Analysis:

Since generators are parallel to the z-axis, the coordinate $z$ is entirely free, yielding $x^2 + y^2 = a^2$.

The enveloping cylinder of the sphere $x^2 + y^2 + z^2 = a^2$ with generators parallel to the x-axis has the equation:

A $y^2 + z^2 = a^2$

B $x^2 + y^2 = a^2$

C $x^2 + z^2 = a^2$

D $x^2 + y^2 + z^2 = 2a^2$

Derivation & Analysis:

Generators parallel to the x-axis mean $x$ is eliminated or free in the tangent cylinder projection, resulting in $y^2 + z^2 = a^2$.

The degree of the homogeneous equation representing a general cylinder in 3D Cartesian coordinates is:

A 1

B 2

C 3

D Depends on guiding curve

Derivation & Analysis:

A general cylinder formed by second-degree surfaces or lines results in a second-degree equation in $x, y, z$.

A cylinder whose generators are perpendicular to its guiding plane is called:

A Oblique cylinder

B Right cylinder

C Enveloping cylinder

D Cone

Derivation & Analysis:

By definition, a cylinder whose generatrix is perpendicular to the plane of the base/guiding curve is a right cylinder.

The number of arbitrary constants in the general equation of a right circular cylinder is:

A 5

B 6

C 7

D 4

Derivation & Analysis:

Determined by direction ratios of axis (2 independent), point on axis (2 independent), radius (1), and orientation constraints.

If $l, m, n$ are direction cosines of the axis of a right circular cylinder, then the relation satisfied by them is:

A $l + m + n = 1$

B $l^2 + m^2 + n^2 = 1$

C $lm + mn + nl = 0$

D $l = m = n$

Derivation & Analysis:

The fundamental property of direction cosines in 3D geometry is that the sum of their squares equals unity ($l^2 + m^2 + n^2 = 1$).

Q10

The projection of a cylinder on a plane parallel to its axis is:

A A circle

B A pair of parallel lines

C A point

D An ellipse

Derivation & Analysis:

Projecting parallel lines along the generator direction onto a parallel plane yields a pair of parallel bounding lines.

Common questions

A sphere has equal coefficients of $x^2, y^2, z^2$ and no product terms ($xy, yz, zx$). A right circular cylinder has equal coefficients of squared terms with specific discriminant relations among product terms. A cone is a purely homogeneous second-degree equation in $x, y, z$.

Failing to normalize direction ratios $(l, m, n)$ to direction cosines when computing perpendicular distances, leading to incorrect radius calculations.

Memorize standard equations where axes align with coordinate axes and use symmetry properties to eliminate incorrect options instantly.